Method and system for optimizing road traffic control in the presence of incidents

ABSTRACT

A method for adjusting control parameters of a traffic management system in a presence of one or more incidents on a network includes junctions in a tree format, and at a lowest level of a tree comprise a group of associated junctions using coordinated timing, wherein each junction is associated with a weight as a function of its height within its tree, and wherein a junction with a highest weight in each tree comprises a lead junction for receiving timing information for its group of associated junctions.

This application is a Continuation application of U.S. patentapplication Ser. No. 15/053,382, filed on Feb. 25, 2016, which is aContinuation application of U.S. patent application Ser. No. 14/038,288,filed on Sep. 26, 2013, now U.S. Pat. No. 9,430,942.

DESCRIPTION Background of the Invention Field of the Invention

The present invention generally relates to performing real-time roadtraffic control in the presence of one or more incidents on the network.More specifically, a method represents, on a tree, a prioritizationacross network junctions prior to an incident, elevates theincident-affected junction higher in priority on the tree, dependingupon the severity level of the incident, associates weights with eachjunction as a function of its height in the tree, and solves thereal-time optimization of control parameters using the weights on thejunctions.

Description of the Related Art

Real-time command centers, such as road traffic command centers, railcommand centers, and supply chain command centers, are complexoperations to manage. For example, within the scope of roadtransportation management and operations, authorities worldwide haverecognized the need to improve the intelligence of their real-timecommand centers.

Command centers are most critical when there are incidents that takeplace on the network. A command center needs to be able to respondeffectively to such incidents with very little response time available.

An important component of an effective and rapid response is thedecision support capability that directs command center operators as towhat actions to take when the need arises. The most difficult task forcommand center operators is determining the appropriate network-wideresponse to incidents.

SUMMARY OF THE INVENTION

In view of the foregoing, and other, exemplary problems, drawbacks, anddisadvantages of the conventional systems, it is an exemplary feature ofthe present invention to provide a structure (and method) for performingreal-time road traffic control in the presence of one or more incidentson the network.

An exemplary feature of the present invention is to provide a real-timetraffic control method and system that leverages data that is currentlycommonly available for most transportation networks.

In a first exemplary aspect of the present invention, to achieve theabove features and objects, described herein is a method for adjustingcontrol parameters of a traffic management system in a presence of oneor more incidents on a network, including: representing, using a treeformat, a prioritization across network junctions prior to an incident,as executed by a processor on a computer; elevating an incident-affectedjunction higher in priority on the tree, depending upon a severity levelof the incident; associating weights with each junction as a function ofits height in the tree; and solving a real-time optimization of thecontrol parameters for the network, using the weights on the junctions.

In a second exemplary aspect, described herein is an apparatus includinga central processing unit (CPU); and a memory, wherein the memory hastangibly embodied thereon a set of machine-readable instructions forexecuting the method described above for adjusting control parameters ofa traffic management system in a presence of one or more incidents on anetwork.

In a third exemplary aspect, also described herein is a method oftraffic control, including representing a network for traffic ascomprising a plurality of links interconnected by a plurality of nodes;loading a traffic model for execution by a processor on a computer, thetraffic model comprising a fluid model, that describes a flow of trafficwithin links of said network, and a junction model, that describes howtraffic on an incoming link to a node is propagated to outgoing links ofthe node, the traffic model assigning priority weights to the nodes ofthe network and using the priority weights for calculating parametersfor optimal traffic control in the network; receiving, as input datainto the computer, indication of current traffic on the network;exercising the traffic model with the received input data to calculateone or more control parameters; and outputting the one or more controlparameters from the computer to one or more devices in the network thatcontrol traffic flow, wherein, upon occurrence of an incident thatcauses a disruption in an existing traffic flow on the network, thetraffic model adapts to the incident by changing one or more of thepriority weights of node within the network, depending upon a locationin the network and a severity of the incident.

The present invention thus provides information that permits a farsuperior response to incident, notoriously difficult because of the needfor split-second decision-making under pressure. The method of thepresent invention can be applied to any network having traffic that iscontrollable and for which there is data sufficient to reasonablydetermine amounts of traffic throughout the network.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other purposes, aspects and advantages will be betterunderstood from the following detailed description of a preferredembodiment of the invention with reference to the drawings, in which:

FIG. 1 shows in block diagram format 100 the input/outputs of anexemplary embodiment of the present invention;

FIG. 2 shows an example of a tree representation 200 of a normal trafficnetwork segment that demonstrates the control mechanism for green wavesof traffic in the network;

FIG. 3 shows a modification 300 of the tree representation following anincident within the network;

FIG. 4 shows a graph 400 exemplarily demonstrating the relationshipbetween flow and density on a link in a traffic network;

FIG. 5 shows an exemplary demonstration 500 of a soft control mechanismusing variable message signs 501;

FIG. 6 illustrates an exemplary hardware/information handling system 600for incorporating the present invention therein; and

FIG. 7 illustrates non-transitory storage medium 700 (e.g., storagemedium) for storing steps of a program of a method according to thepresent invention.

DETAILED DESCRIPTION OF AN EXEMPLARY EMBODIMENT OF THE INVENTION

Referring now to the drawings, and more particularly to FIGS. 1-7, anexemplary embodiment of the method and structures according to thepresent invention will now be described.

The present invention performs real-time traffic control in the presenceof one or more incidents on a network. Although the present invention isdescribed herein relative to transportation networks, it is noted thatthis inventive method is not intended as being limited to this specifictype of network or system. It can be applied to other types of networksother than transportation networks, such as, for example, water orenergy supply systems, that may, however, be less well tracked andequipped with real-time data provisions than traffic transportationnetworks. Some traffic transportation networks have recently evolved toprovide real-time data even as incidents occur.

In the context of the present invention, the term “network” is intendedas referring to combination of a plurality of links upon which trafficcan flow from one point of the network to another point in the network.The links are interconnected by junctions (nodes), meaning points in thenetwork at which links interconnect in a manner that traffic leaves onelink and enters into at least one other link in the network, toselectively change direction of movement within the network. In thecontext of the present invention, an “incident” is intended as referringto any event that causes a disruption of traffic flow in at least onelink or node of the network relative to the traffic flow immediatelyprior to the occurrence of the event.

As exemplarily shown in the block diagram 100 of FIG. 1, the method ofthe present invention is exemplarily implemented as a Control PlanOptimizer (CPO) module 101 that is part of a larger tool 102 thatprovides a set of input functionalities, including the CPO module of thepresent invention, from incident detection through incident durationprediction and incident impact propagation. The CPO function of thepresent invention is intended to provide optimal response plans tomitigate negative impacts of one or more incidents on a road network andis triggered by the detection of at least one incident having sufficientimpact to the network performance.

As an overview, the CPO module 101 accepts as inputs a set ofnetwork-wide traffic predictions 103 and, in particular, on the networklink having a detected traffic incident 104, as well as the non-impactednetwork links. The traffic predictions are assumed to include flowpredictions as well as occupancy predictions, and/or speed predictions.Another input to the CPO model is a prediction of the duration of theincident(s) 105, available either from historical data on similarincidents, by user input, or as deduced from speed or occupancypredictions of the incident link, using a predicted return-to-normaltime. Thus, the CPO module receives current network traffic data anduses data on current incidents, including their estimated severity andtheir predicted duration, in conjunction with the current data 106 fortraffic on the network at the time of an incident.

An incident can be automatically detected by the system controllermodule, based on changes of the current traffic data, or an incident canbe manually input by an operator. The method leverages simulated trafficflow data to take into account likely routes used by drivers, in orderto identify locations on the network that should be subject to optimizedcontrol in the presence of incidents.

Thus, the approach of the present invention takes a network-wide view ofthe traffic control problems, to take into account effects of thecontrol actions on the full traffic state. This is a considerableimprovement over the current state of traffic control. The completemathematical model used to implement this mechanism is included afterthe discussion below that summarizes an exemplary underlying mechanismof the present invention.

A key mechanism used in one exemplary embodiment of the presentinvention, exemplarily shown in FIG. 2, is that a hierarchy of junctionsis maintained across the junctions on the road network, using a treerepresentation 200. Using that hierarchy, the tool can then establishand maintain network timing parameters to enable so-called “green waves”of traffic flow. When an incident occurs in the network, the timingparameters for lights in the system can be adjusted, to adapt to theincident, including possibly adapting the existing green waves in thenetwork to work around the deviations caused by the incident.

FIG. 2 exemplarily shows two separate green waves occurring in thenetwork. The first green wave involves junctions 1, 3, 8, 2, 5, 9 anduses junction 8 as the lead junction for that green wave. The secondgreen wave involves junctions 4, 6, 7 and uses junction 4 as the leadjunction. In other words, in this example, it is the timing of junctions8 and 4 that are used by the other signals/junctions in the group, withthe appropriate offset, meaning the time to get from one junction to thenext, assuming normal traffic speeds at the current time.

FIG. 3 shows exemplarily an incident 302 having occurred betweenjunctions 3 and 8, and the resultant escalation of junction 3 asbecoming more important at that location in the network due to theincident. Once an incident occurs which blocks partially or totally theaccess to junction 8, the present invention can increase the priority ofthe junction 3 upstream of 8, as indicated by label 301, so that whenthe optimal traffic signal timings are computed, with a goal ofmaximizing the network flow (e.g., throughput) through the network, agreater emphasis will be placed on the flow through junction 3, e.g., tothe right or the left, since the other links in the small figure next tothe tree are not shown in the figure, but there would typically be otherlinks going out of 3 besides the link to 8.

It should be clear that an incident can occur anywhere within a network,at either a junction or along a link. FIG. 4 exemplarily shows therelation 400 between the flow and the density at a junction in thenetwork, and how the tool of the present invention can model the impactof the control parameter, which is the green time at the signal. Inessence, reducing this green time lowers the maximum flow that can goout of a link, which is, in fact, reducing the top of the triangle 401to the trapezoid 402. That is, in the model described in detail below,the control parameters (g,h) are optimized a posteriori, once anincident becomes known or recognized. Once an incident has occurred, itcan thereafter serve as historical data that can provide initialconditions for optimization when similar incidents occur, not only inthe context of the present invention dealing with incident impactminimization of traffic disruption, but also in the context of makingpredictions for future incidents in the network, such as predictingdurations of similar incidents and predicting impacts on traffic flowfor similar incidents.

Thus, for example, co-pending application entitled “Prediction of Localand Network-Wide Impact of Non-Recurrent Events in TransportationNetworks”, could use traffic data related to a current incident asfuture historical incident data, using that historical data to thenprovide the input data shown in FIG. 1 of the present application fortraffic prediction 103 and for incident duration prediction 105 into theCPO module 101 of the present invention.

The illustration 500 in the FIG. 5 shows that a variable message sign501 can also be provided as an output of the tool of the presentinvention, as a secondary means of traffic control, by giving a messageto drivers to exit towards a different road. The model determines thepercentage of flow, within limits of the compliance rate (e.g., seelabel 502 in FIG. 5) of drivers to messages, that should be diverted.This translates to how long the sign should display the message, forexample.

The following sections present details of the mathematical theformulation and approach underlying the CPO component of the presentinvention.

Preliminaries

We define a graph representing the full traffic network of study, G=(V,E), where V is the set of nodes in the network, |V| its cardinality, andE the set of links, or edges, connecting the nodes. Let |E|=m. The graphG is assumed to be strongly connected. Each edge e∈E is directed from atail node, tail (e)∈V, to a head node head (e)∈V. For each node v∈V, wedefine the sets of outgoing and incoming edges from/to that node as:O(v): ={e∈E tail (e)=v} and I(v):={e∈E|head (e)=v}. A subset {tilde over(V)} of the junctions V are assumed to be controllable. That is, {tildeover (V)}⊂V is the set of junctions having at least one controllablemechanism controlled by a computer executing the method of the presentinvention.

In an exemplary embodiment, traffic is represented in the presentinvention using a pair of models:

-   -   1. a macroscopic fluid model describes the flow of traffic        within each link of the network; and    -   2. a junction model, defined on each node of the network,        describes how the traffic on an incoming link to a node is        propagated to the outgoing links of the node.

Using a macroscopic fluid model implies that each link is assumed to bea homogeneous entity with length L_(e). The traffic is characterized byits density, itself a function of space and time, i.e., ρ(x,t). Trafficflow is in turn a concave function of density, Q(ρ(x,t)), known as theso-called “fundamental diagram”, and defined over the interval ρ∈[0,ρ_(max,e)] for some non-negative, possibly link-specific, maximaldensity value, ρ_(max,e). The density at which Q(⋅) takes its maximum islabeled the critical density, ρ_(c,e), and may be specific to each edge,e; the flow at the critical density is called Q_(max,e)=Q(ρ_(c,e)). Fortraffic data defined in terms of occupancy, or speed, suitabletransformations of the density, or flow, respectively, are employed.

It is noted that FIG. 4 provides a simplistic example of this concept offundamental diagrams.

The model is by nature dynamic and, hence, the parameters of the modelare defined over a time horizon of discrete time periods n∈[0,N]. It isassumed that nodes are not able to store traffic. Therefore, at eachnode, the flow is propagated from the incoming links into the node tothe outgoing links. If e is an incoming link at a node v and e anoutgoing link, and if the splitting rates of traffic at the nodes areknown, then p_(ef) ^(n) is the, possibly time-varying, value indicatingthe proportion of the outgoing flow from link e that would like to enterlink f, with Σ_(l∈O(v))p_(e,l) ^(n)=1 for every e∈I(v), v∈V, because ofthe lack of storage capacity at the nodes.

A primary objective of the method of the present invention is todetermine optimal control plans in the presence of a detected incident.When an incident is detected at time τ, a prediction of the trafficlevel (flow, speed, and/or occupancy) must be computed on each impactedlink and for each subsequent time step in a pre-defined forecastinghorizon [n+1, n+τΔτ], using the traffic model described in greaterdetail below. However, in addition to the traffic model presented here,it is assumed that traffic predictions are available providing flow(and/or occupancy, speed) forecasts on a subset Ê⊆E of the non-impactedlinks of the network, as well as on the incident link itself, at apre-determined number τ of discrete Δτ-length time steps into the futurefrom the current time, n, [n+1,n+τΔτ]. These external predictions willtherefore be used as boundary conditions on the model defined here.

As briefly mentioned above, the pre-defined forecasting horizon can bederived in any of several ways. For example, historic data on incidentscan be used to derive a generic value used for all incidents or thehistoric data can be used to derive different values of forecastinghorizons for different types of incidents or incidents in differentlocations in the network. Another possible strategy is to initially usea pre-defined forecasting horizon and then monitor on-going traffic onthe network, to determine whether this forecasting horizon seemssufficient in length to accommodate the current incident, and thenupdating the forecasting horizon as required, based on determiningwhether traffic has recovered from the incident. This approach issimilar to the method described in co-pending application entitled“Prediction of Local and Network-Wide Impact of Non-Recurrent Events inTransportation Networks”, the contents of which of hereby incorporatedby reference.

Two forms of control are available:

-   -   (1) traffic signal green time changes, given by the decision        variable t_(vj) ^(n), for phase j (which collects a number of        link-link movements that may take place at the same time) at        junction v during time step n; and    -   (2) traveler information provision, such as setting variable        message sign (VMS) content, e.g., see label 501 in FIG. 5, given        by the decision variable h_(ef) ^(n)∈[0,1] on link e,        representing the proportion of traffic on link e suggested to        divert at link f at time n. Since traveler information is        considered a soft control where user compliance is optional, we        define also a compliance rate ξ_(ef) ^(n), both time- and        location-varying, in that the compliance rate may depend on the        spatial separation between link e and link f. Label 502 of FIG.        5 demonstrates visually this compliance rate ξ, as shown by the        dotted line demonstrating the change in direction at the next        intersection.

For both types of control, we shall assume that control changes may onlybe made at time points n mod Δτ.

The Traffic Model

The traffic model is comprised of a dynamic link flow model and ajunction model along with a set of boundary conditions. The models aretypically solved sequentially over the set of links and nodes in thenetwork.

The link model relies on the resolution of a kinematic wave equation topropagate and conserve the flow of traffic across the link in thetime-space plane. The equation in this case, defined in terms of trafficdensity, states that

$\begin{matrix}{{{\rho_{t}\left( {x,t} \right)} + {\frac{d\;{Q\left( {\rho\left( {x,t} \right)} \right)}}{d\;\rho}{\rho_{x}\left( {x,t} \right)}}} = 0} & (1)\end{matrix}$where dQ(ρ(x,t))/dρ is the derivative of the flow function of thefundamental diagram, and where the definition of the derivative ispiecewise when Q(⋅) is piecewise continuously differentiable.

The resolution of each link model involves a spatial discretization ofthe link into a finite number of discrete cells, which satisfy thekinematic wave equation (1) across their boundaries. Let the number ofcells on link e be given by C_(e) and the number of cells across thenetwork given by C=Σ_(e∈E)C_(e). In addition, a fine time discretizationis needed. For each prediction time step 1 . . . τ described above, anumber of smaller time steps are required to solve the traffic modelover each link. For simplicity, we refer to each smaller time step inthe following as n. The two time scales need only be distinguished inthe control model described in more detail in the next section entitled“The Control Model”.

The network structure is defined in terms of cells as it is in terms oflinks so that head (i)=v means that the downstream end of cell i is thenode v, and I(v) may refer either to the set of incoming network linksor cells adjacent to node v, depending on the context. Letting i, i+1denote the upstream and downstream cells, respectively, the flow Q isgiven by the fundamental diagram evaluated at the following densityvalues:

$\begin{matrix}{Q_{i,e}^{n} = \left\{ \begin{matrix}{{Q\left( \rho_{i}^{n} \right)},\mspace{14mu}{{if}\mspace{14mu}\left( {\rho_{i + 1}^{n} \leq \rho_{i}^{n} \leq \rho_{c}} \right)}} \\{{Q\left( \rho_{i + 1}^{n} \right)},{{if}\mspace{20mu}\left( {\rho_{c} \leq \rho_{i + 1}^{n} \leq \rho_{i}^{n}} \right)}} \\{Q_{\max,e}^{n},\mspace{14mu}{{if}\mspace{14mu}\left( {\rho_{i + 1}^{n} \leq \rho_{c} \leq \rho_{i}^{n}} \right)}} \\{{\min\left\{ {{Q\left( \rho_{i}^{n} \right)},{Q\left( \rho_{i + 1}^{n} \right)}} \right\}},\mspace{14mu}{{if}\mspace{14mu}\left( {\rho_{i}^{n} \leq \rho_{i + 1}^{n}} \right)}}\end{matrix} \right.} & (2)\end{matrix}$where cell i lies on network link e. Q_(max,e) represents, in general,the value Q(ρ_(c,e)).

Expressing flow conservation for each cell leads to a temporal update ofthe density on a link according to:

$\begin{matrix}{{\rho_{i}^{n + 1} = {\rho_{i}^{n} + {\frac{\Delta\; N}{\Delta\; x}\left( {{Q\left( \rho_{i}^{n} \right)} - {Q\left( \rho_{i + 1}^{n} \right)}} \right)}}},} & (3)\end{matrix}$where ΔN is the duration of the time interval and Δx the length of thecell.

At the junctions, the flow from each incoming cell into the node must bepropagated to each outgoing link. There are clearly an infinite numberof ways that flow may be allocated across each junction. However, byfixing the splitting rates at each node n, to a set of fixed values{p_(ef) ^(n)|e∈I(v), f∈O(v)}, the range of possible flow allocationsacross the junctions is considerably reduced. Specifically, it issupposed that the amount of flow that is able to enter the junctiongiven upstream and downstream constraints is split in proportion to thefixed splitting rates of B_(v) for each node v. Using this approach, alinear program can be expressed for each junction which serves tocompute the maximal flow that can be sent from the incoming edges at thenode to the outgoing edges, subject to the constraints given by the cellflows and capacities, for each time interval n. Letting B_(v) ^(n)∈

₊ ^(|O|×|I|) be the node-specific matrix of splitting rates, i.e.,B_(ef,v) ^(n)=p_(ef) ^(n) for head (e)=tail(f)=v, the possiblynon-unique junction flows into node v are given by the solutionq^(*,n)∈R₊ ^(|I(v)|) to:

$\max\limits_{\{ q_{e}\}}{\sum\limits_{e \in \;{I{(v)}}}\; q_{e}}$Such that B _(v) q≤R _(f) ^(n) ,f∈O(v),  (4)0≤q≤S _(e) ^(n) ,e∈I(v),where S_(e) ^(n) gives the maximal amount of flow that can be sent fromlink e through node v at time step n, and R_(f) ^(n) the maximal amountthat can be received into link f from the node, defined as thehypocritical and hypercritical portions of the fundamental diagram,respectively. Specifically, S_(e) ^(n) and R_(f) ^(n) are defined overthe interval [0, ρ_(max)] with S_(e) ^(n) taking the values of Q_(e)(⋅)for e∈I(v) in the hypocritical region and taking the value Q_(max,v)^(n), for ρ_(e)∈[ρ_(c,e),ρ_(max,e)]:

$\begin{matrix}{S_{e}^{n} = \left\{ {\begin{matrix}{{Q\left( \rho_{e}^{n} \right)},} & {{if}\left( {\rho_{e} \in \left\lbrack {0,\rho_{c,e}} \right\rbrack} \right)} \\{Q_{\max,e}^{n},} & {{if}\left( {\rho_{f} \in \left\lbrack {\rho_{c,e,}\rho_{\max,e}} \right\rbrack} \right)} \\{0,} & {otherwise}\end{matrix},} \right.} & (5)\end{matrix}$defined as such to permit subsequent optimization of the traffic signalcontrol variable, and similarly R_(f) ^(n) set to Q_(f)(⋅) for f∈O(v) inthe hypercritical region and taking the value Q_(max,f) ^(n) forρ_(f)∈[0, ρ_(c,f)]:

$\begin{matrix}{R_{f}^{n}\left\{ \begin{matrix}{Q_{\max,f}^{n},} & {{if}\left( {\rho_{f} \in \left\lbrack {0,\rho_{c,f}} \right\rbrack} \right)} \\{{Q\left( \rho_{f}^{n} \right)},} & {{if}\left( {\rho_{f} \in \left\lbrack {\rho_{c,f},\rho_{\max,e}} \right\rbrack} \right)} \\{0,} & {otherwise}\end{matrix} \right.} & (6)\end{matrix}$The junction flows out of the node are then given by B_(v) ^(n)q^(*,n).

In order to accurately model the impact of an incident on the trafficflow propagation, the boundary conditions of the traffic model must takeinto account the predicted state of the network at the spatialboundaries of the impacted region, for each time step. Given thejunction model defined above, the definitions of receiving and sendingflows can be used as a proxy for the incident-induced boundaryconditions. In other words, the spatial boundaries of the traffic modelshould include precisely the set of impacted links, augmented by the setof incoming links at each upstream boundary node, and the set ofoutgoing links at each downstream boundary node. Note that this set ofincoming and outgoing links at boundary nodes may contain duplicateentries due to the presence of cycles in the network. The effects ofsuch dependent flows will be neglected at the present time.

The Control Model

The traffic model described above provides the predicted traffic flow onthe incident-impacted links over the time horizon, N, as a function ofthe control variables, t and h. The model is defined so that the initialtime corresponds to the current time, as in a rolling horizon or a modelpredictive control (MPC) approach. It is necessary to distinguish herethe two different time scales: the flow model described in the previoussection requires a spatio-temporal discretization into cells and timesteps such that the minimum time required for a vehicle to traverse eachcell is no less than the time discretization interval. We shall refer tothis time index, as before, as n=1, 2, . . . N. On the other hand, theoptimal controls should be updated less frequently, i.e., at the subsetof the n time instants, given by the τ control decision points. Inpractice, only the first, or the first few, optimal controls areimplemented, the remainder of the time horizon is used to approximate aclosed feedback loop.

The solution to the control model should increase the throughput in theaffected region without having a negative impact to the boundary of theregion. Typical objectives include the minimization of total networkdelay, defined as a weighted sum of the flows on the links of thenetwork, and the sum of squared occupancies or normalized densities.

The junction model of equation (4) above involves the maximization ofthe throughput, or flow through the junction. Given that flow ismaximized in the junction model, utilizing a minimization objective oftotal delay, itself a weighted sum of flow, would be incompatible withthis formulation. Thus we shall seek the maximization of the outflows onthe network whilst, as in the junction models above, maximizing the sumof the flows at all of the junctions, the latter objective having alower weight.

In addition to including the impact of the incident through thespatio-temporal boundary conditions, the effect of traffic controlmeasures on the flow must also be represented in the traffic model. Weconsider primarily two forms of traffic control: hard controls,accomplished via traffic signal timing changes, referred to as t, andsoft controls, defined as driver information and recommendations, h,displayed to drivers on variable message signs at fixed locations, on awebsite, or using some form of voice-based text messaging.

Various mechanisms are available to incorporate the effect of controlsinto the traffic model: the shape of the fundamental diagram,Q_(e)(⋅)=Q_(e)(⋅, Q_(max,e)), for each link, e∈I(v), and morespecifically, for the last cell within that link, entering the junction,may be modified according to a pre-defined function of its parameterQ_(max,e) given the traffic signal timing change at a junction v.Specifically, for a network link, e, and cell C_(e), where head (e)=head(i), and there is only a single traffic signal approach is controlled,for instance, the approach from link e to f,Q _(max,e)(g _(e) ^(n))=Q _(max,e) ^(n) g _(e) ^(n),  (7)where g_(e) ^(n) is the reduction from the total design capacityQ_(max,e) ^(n) due to traffic signal red time at the downstream junctionof link e.

The impact of this modification on the junction would be through thesending functions, so that S_(e)=S_(e)(g_(e) ^(n)), and on the last cellon the link, via the modified maximal capacity of equation (7) in thefundamental diagram, Q_(max,e) ^(n) of equation (2). The impact of thecapacity reduction at the junction on the rest of the cells on the linkis handled via the traffic model through flow propagation. In this way,the impact of the traffic signal change acts an average level over thetraffic cycle, by reducing (or increasing) the maximal outflow on thecells entering the junction as a function of a percent reduction (orincrease) in green time.

Note also that we do not explicitly consider phases here, the green timechange on all the phases would be adjusted and the impact of that totaladjustment on the green times of all phases on the other links enteringthe junction adjusted accordingly, as follows.

The variable g_(e) ^(n) is, in practice, a function of the timeallocated to each phase of the traffic signal cycle at the junctionv=head (e), denoted t_(vj) ^(n) for the jth phase at the junction vduring control time step n, where H_(v) is the set of the phasesavailable at junction v. Then g_(e) ^(n) for link e is given by the sumof the green time allocated to vehicles coming from link e into thejunction over all phases, divided by the total cycle length, Lv:

$\begin{matrix}{{g_{e}^{n} = {\frac{1}{L_{v}}{\sum\limits_{j \in H_{v}}\;{t_{vj}^{n}{\sum\limits_{{f \in {j{(1)}}}:{e \in {j{(0)}}}}\;\frac{\eta_{ef}}{Q_{\max,e}}}}}}},} & (8)\end{matrix}$where j(0) is the vector of the origin link of each movement of thephase j in H_(v) and j(1) is the vector of destination links of eachmovement, and η_(ef) ^(j) is the portion of the capacity of link epertaining to the movement ef, such as the capacity of a turning lane,if the movement ef is a turning movement. Whereas the set of phases andthe cycle length typically vary, e.g., for the morning versus theevening or night, we assume that they are constant for the duration ofthe optimization, and hence do not require an index. The decisionvariables are therefore the green time allocated to each signal phase,t_(vj) ^(n), for each phase j and each junction v, at every control timestep n. In addition, each signal phase variable is associated with boundconstraints and a simplex-type constraint on the total cycle time, forevery time step n and every junction v and phase j:

$\begin{matrix}{{l_{vj} \leq t_{vj}^{n} \leq u_{vj}},} & (9) \\{{\sum\limits_{j \in H_{v}}\; t_{vj}^{n}} = {L_{v}.}} & (10)\end{matrix}$

The matrix B_(v) of splitting rates at junction v can be defined interms of the t_(vj) ^(n) for controllable junctions. Let the set {tildeover (V)}^(n) be all such controllable junctions at time n. Similarly,the set of links entering junctions {tilde over (V)} is referred to as{tilde over (E)}^(n) at time n. Specifically, for v∈{tilde over (V)},the splitting rate for an adjacent pair of links (e,f) is the capacityassociated with a given movement over all phases that include thatmovement multiplied by the total time the movement is active, normalizedby the same over all movements:

$\begin{matrix}{{{p_{ef}^{n}\left( t_{v}^{n} \right)} = \frac{\sum\limits_{j \in {H_{v}:{{({e,f})} \in j}}}\;{t_{vj}^{n}\eta_{ef}}}{\sum\limits_{j \in H_{v}}{t_{vj}^{n}{\sum\limits_{{({e,f})} \in j}\;\eta_{ef}}}}},} & (11)\end{matrix}$for each pair (e,f) such that head(e)=tail(f)=v. Note that p_(ef)^(n)(t_(v) ^(n)) is nonlinear in the control variable t_(v) ^(n).

For links where traffic information provision is available, e.g., avariable message sign is present, h_(ef) ^(n) is introduced into thetraffic model by adjusting the splitting rates, p_(ef) ^(n) by aquantity to optimize, dp_(ef) ^(n), and thus the decision variable h^(n)enters the traffic model through the node-specific matrices B_(v)(t_(v)^(n),h^(n)), which depend on the traffic signal control variable as wellas via equation (11):

$\begin{matrix}{{{{dp}_{ef}\left( h^{n} \right)} = {\xi_{ef}^{n}h_{ef}^{n}}},} & (12) \\{{{p_{ef}^{n}\left( {t_{v}^{n},h^{n}} \right)} = {{p_{ef}^{n}\left( t_{v}^{n} \right)} + {{dp}_{ef}^{n}\left( h_{ef}^{n} \right)}}},} & (13) \\{{{- \frac{p_{ef}^{n}\left( t_{v}^{n} \right)}{\xi_{ef}^{n}}} \leq h_{ef}^{n} \leq \frac{1 - {p_{ef}^{n}\left( t_{v}^{n} \right)}}{\xi_{ef}^{n}}},} & (14)\end{matrix}$where ξ_(ef) ^(n) is a given constant quantifying the compliance rateξ_(ef) ^(n) for the adjacent link pair (e,f) at time step n. The factthat the compliance rate depends upon both e and f. allows for a highercompliance when the diversion concerns the adjacent network link, i.e.,head(e)=tail(f), since the cause of the suggested diversion may bevisible to the driver or for diversions that are more likely to befollowed by drivers. For diversions suggested further upstream,compliance rates may be lower. The constraint (14) provides the 0-1bounds on the total splitting rates for each movement, p_(ef)^(n)+dp_(ef) ^(n). Note, however, that the constraints are nonlinear forjunctions v∈{tilde over (V)} where t_(v) ^(n) is a decision variable.

The formula 15 below sets the splitting rates of those link pairs with anon-zero soft control, and then re-allocates the remaining proportion tothe outgoing links p_(el) ^(n), l∈O(head(e)), h_(el)=0 whose travelerinformation control variables are null, while maintaining theproportionality of the initial splitting percentages for those outgoinglinks.

                                           (15)${p_{ef}\left( {t_{v}^{n},h^{n}} \right)} = \left\{ \begin{matrix}{{{p_{ef}^{n}\left( t_{v}^{n} \right)} + {\xi_{ef}^{n}h_{ef}^{n}}},} & {{{if}\mspace{11mu}\left( {h_{ef}^{n} \neq 0} \right)},{and}} \\{{{p_{ef}^{n}\left( t_{v}^{n} \right)} + \frac{{p_{ef}^{n}\left( t_{v}^{n} \right)}\left( {1 - {\sum\limits_{f^{\prime} \in {O{({{head}{(e)}})}}}\;{p_{ef}^{n}\left( t_{v}^{n} \right)}} + {\xi_{ef}^{n}h_{{ef}^{\prime}}^{n}}} \right)}{\sum\limits_{{l \in {O{({{head}{(e)}})}}},{h_{el} = 0}}\;{p_{el}^{n}\left( t_{v}^{n} \right)}}},} & {{if}\mspace{14mu}{\left( {h_{ef}^{n} = 0} \right).}}\end{matrix} \right.$

Formula 15 holds for non-adjacent links as well as adjacent links, forexample, h_(ue) ^(n), where link u is upstream from and non-adjacent tolink e and link e is directly adjacent to link f. However, this requiresa mechanism for identifying which downstream link e is the relevantlink, since there may be more than one path from link u to link f, as invia links e₁ and e₂. In addition, given the complexity of the model, andthe non-uniqueness of the optimal solution, it is preferable to optimizethe travel information control variable h_(ef) ^(n) for adjacent linkse,f only and to solve a separate problem to distribute the travelerinformation on multiple upstream links in a way that achieves the targetdiversion rate of p_(ef) ^(n)+ξ_(ef) ^(n)h_(ef) ^(n).

Nonlinear Control Model

A nonlinear formulation of the optimal control model using the objectivecriterion described above, with 0≤ζ_(ef)<<1, the weighting factorbetween the exiting flows and the full set of network flows, may beexpressed as follows. Let q_(C) _(e) _(,e) be the flow out of the lastcell on link e into the junction, such that the node v=head (e) isadjacent to the cell. The flows into the cells i on the link e arelabeled q_(ie) for i=0 . . . C_(e)−1. The flow into the first cell onlink e, that is, the cell exiting the upstream junction v=tail(e), isreferred to as q_(0,e), and does not have a cell immediately upstream soits flow is computed using the junction outflows. Q_(i,e-1)(ρ^(n),g^(n)) is given by equation (2). Recall that the control optimizationframework is considered to be a Model Predictive Control (MPC) thatsolves the time-dependent control problem on a rolling-horizonframework, implementing only the first control point and then updatingthe time steps and solving again. For notational simplicity, supposethat the current time is however always denoted by n=1. Then, we have,

$\begin{matrix}{\max\limits_{\rho,q,t,h}{\sum\limits_{n = 1}^{N}\;\left( {{\sum\limits_{e \in E}\;{g_{e}^{n}q_{C_{e},e}^{n}}} + {\sum\limits_{i = 0}^{C_{e} - 1}\;{ϛ_{ei}q_{i,e}^{n}}}} \right)}} & \left( {16a} \right)\end{matrix}$Such that:

$\begin{matrix}{{\rho_{j,e}^{n + 1} = {\rho_{j,e}^{n} + {\frac{\Delta\; N}{\Delta\; x}\left( {q_{j,e}^{n} - q_{{j - 1},e}^{n}} \right)}}},{j = {1\mspace{14mu}\ldots\mspace{14mu} C_{e}}},{e \in E},{n = {1\mspace{11mu}\ldots\mspace{14mu} N}}} & \left( {16b} \right) \\{{q_{0,e}^{n} = {\sum\limits_{j \in {I{({{tail}{(e)}})}}}{{\rho_{j,e}\left( {t_{v}^{n},h^{n}} \right)}q_{{Cj},j}^{n}}}},{e \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {16c} \right) \\{{g_{e}^{n} = {\frac{1}{L_{v}}{\sum\limits_{j \in H_{v}}{t_{vj}^{n}{\sum\limits_{{f \in {j{(1)}}}:{e \in {j{(0)}}}}\frac{\eta_{ef}}{Q_{\max,e}}}}}}},{e \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {16d} \right) \\{{h_{ef}^{n + 1} = h_{ef}^{n}},{{\left( {n + 1} \right)\;{mod}\;\Delta\;\tau} \neq 0},{f \in O},{f \in {O\left( {{head}(e)} \right)}},{e \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {16e} \right) \\{{g_{e}^{n + 1} = g_{e}^{n}},{{\left( {n + 1} \right)\;{mod}\;\Delta\;\tau} \neq 0},{f \in {O\left( {{head}(e)} \right)}},{e \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {16f} \right) \\{{q_{{i + 1},e}^{n} = {Q_{i,e}\left( \rho^{n} \right)}},{i = {{0\mspace{14mu}\ldots\mspace{14mu} C_{e}} - 2}},{e \in E},{v = {{head}(e)}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {16g} \right) \\{{q_{C_{e},e}^{n} = {Q_{C_{e},e}\left( {\rho^{n},g_{e}^{n}} \right)}},{e \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {16h} \right) \\{{{\sum\limits_{j \in H_{v}}t_{vj}^{n}} = L_{v}},{v \in \overset{\sim}{V}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {16i} \right) \\{{{\sum\limits_{e \in {I{(v)}}}{{B_{{ef},v}\left( {p\left( {t_{v}^{n},h^{n}} \right)} \right)}q_{C_{e},e}^{n}}} \leq R_{f}^{n}},{f \in {O(v)}},{e \in E},{v = {{head}(e)}},{w = {{head}(f)}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {16j} \right) \\{{0 \leq q_{C_{e},e}^{n} \leq {S_{e}\left( g_{e}^{n} \right)}},{e \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {16k} \right) \\{{l_{vj} \leq t_{vj}^{n} \leq u_{vj}},{j \in H_{v}},{v \in \overset{\sim}{V}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {16l} \right) \\{{{- \frac{p_{ef}^{n}\left( t_{v}^{n} \right)}{\xi_{ef}^{n}}} \leq h_{ef}^{n} \leq \frac{1 - {p_{ef}^{n}\left( t_{v}^{n} \right)}}{\xi_{ef}^{n}}},{f \in {O\left( {{head}(e)} \right)}},{e \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {16m} \right) \\{{0 \leq \rho_{i}^{n} \leq \rho_{\max,e}},{i = 0},{\ldots\mspace{14mu} C_{e}},{e \in E},{v = {1\mspace{14mu}\ldots\mspace{14mu} V}}} & \left( {16n} \right) \\{{q_{i,e}^{n} \geq 0},{i = 1},{\ldots\mspace{14mu} C_{e}},{e \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {16o} \right)\end{matrix}$

The objective function is a product of the allocated capacity per linkand the flow on that link and contains two sets of terms since a higherweight is applied to the flows on the cell that exit each link into itsdownstream junction. The other flows are also maximized but with aweight ζ_(ei) much less than one. The equality constraint (16b)describes flow propagation between cells on each link. The constraint(16c) defines the flow into the last cell on a link as a function of thesplitting rate function of (15) and the flow into the junction justupstream of the cell and is therefore nonlinear, since p(h^(n)) islinear in h^(n) and nonlinear in t^(n). Constraint (16d) is adefinitional constraint that defines the effect of the controloptimization of the phase timing at a junction in terms of each upstreamlink and its associated capacity reduction, and is used only to simplifythe notation in the remainder of the model. Constraints (16e) and (16f)state that the control decision variables t^(n) and h^(n) may be updatedon the time scale given by Δτ and not in between. Constraint (16g)states that the fundamental diagram of (2) gives the flows on the cellsinternal to each link, where equation system (2) depends upon thetraffic signal decision variable in as defined in (8); as stated, thisconstraint is non-convex due to the min function. In addition,constraints (16g) and (16h) are given by a piecewise-concave flowfunction. The constraint (16i) is a simplex constraint on the totalcycle time at each controllable junction v∈{tilde over (V)}.

The inequality constraint (16j) provides the flows into the junctionfrom the first cells on the links just upstream of the junction; theleft-hand side contains, therefore, the control variable and both theflow variables, the phase timing variables t_(vj) ^(n), and the trafficinformation decision variables h^(n), via the matrix of splitting rates,B(p(t_(v) ^(n),h^(n))) and hence is nonlinear. The index of thereceiving flow function on the right-hand side of constraint (16j)corresponds to the cell outgoing from the junction n. At the junctions,the sending flows are expressed as a function of the traffic controldecision variable g^(n)(t_(vj) ^(n)) while the receiving flows are notdefined as a function of the upstream signal timing change. Theinequality constraint (16k) is a function of the traffic signal controldecision variable, g^(n), and provides the constraint on the sendingflows into the junction. Note that S_(e) corresponds to the sending flowfunction for the first cell on link e, incident to the junction.Constraints (16l)-(16o) provide the bounds on the control decisionvariables, the density variable, and the flow variables t^(n), h^(n),ρ^(n) and q^(n), respectively.

The model given by (16a)-(16o) is therefore a continuous optimizationproblem with a non convex, nonlinear objective function and a nonconvex,nonlinear feasible region. Based on the MPC approximate closed-loopapproach, it is assumed that the model is re-run at every timestep τ,with the solution t*(Δτ),h*(Δτ) implemented, and then the time index nreset to 1 to re-run the model over the full horizon N. If computationtimes are long relative to the time period given by Δτ, it is possibleto implement the first few optimal t*,h* from the model run at n=1.

Reformulation of the Control Model

By assuming that the fundamental diagram Q_(e)(ρ^(n),Q_(max,e) ^(n)) ispiecewise-linear for each link e and time step, n, the system (2)defining intercell flow dynamics can be simplified. In our setting, theshape of each fundamental diagram depends upon g_(e) ^(n) whenhead(e)∈{tilde over (V)}. Thus, g_(e) can take values so as to reducethe maximum feasible outflow from link e, via a trapezoidal fundamentaldiagram on the link e. In other words, the free-flow speeds ν, backwardwave speeds ω, and ρ_(max) need not change in the case of apiecewise-linear fundamental diagram by the use of g^(n) as a controlparameter acting through Q_(max). What does change in allowing for areduction of Q_(max) through a trapezoidal fundamental diagram is ρ_(c):in this case, there is a range of values for the critical density,ρ_(c)∈[ρ_(c1),ρ_(c2)].

The sending and receiving functions, S and R, in terms of the left andright bounds of the critical densities and in terms of the free flow andbackward wave speeds of the trapezoidal fundamental diagram, are then:

$\begin{matrix}{{S_{e}^{n}\left( g_{{head}{(e)}}^{n} \right)} = \left\{ {\begin{matrix}{{v_{e}\rho_{e}^{n}},} & {{if}\mspace{14mu}\left( {\rho_{e} \in \left\lbrack {0,\rho_{{c\; 1},e}} \right\rbrack} \right)} \\{{Q_{\max,e}^{n}g_{{head}{(e)}}^{h}},} & {{if}\mspace{14mu}\left( {\rho_{e} \in \left\lbrack {\rho_{{c\; 1},e},\rho_{\max,e}} \right\rbrack} \right)} \\{0,} & {otherwise}\end{matrix},} \right.} & (17) \\{and} & \; \\{R_{f}^{n} = \left\{ {\begin{matrix}{Q_{\max,f}^{n},} & {{if}\mspace{14mu}\left( {\rho_{j} \in \left\lbrack {0,\rho_{{c\; 2},f}} \right\rbrack} \right)} \\{{Q_{\max,f}^{n} + {\omega_{f}\left( {\rho_{f}^{n} - \rho_{{c\; 2},f}^{n}} \right)}},} & {{if}\mspace{14mu}\left( {\rho_{f} \in \left\lbrack {\rho_{{c\; 2},f},\rho_{\max,f}} \right\rbrack} \right)} \\0 & {otherwise}\end{matrix},} \right.} & (18) \\{where} & \; \\{{\rho_{{c\; 1},e}^{n} = \frac{Q_{\max,e}^{n}g_{{head}{(e)}}^{n}}{v_{e}}},{e \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & (19) \\{{\rho_{{c\; 2},f}^{n} = {\rho_{\max,f} + \frac{Q_{\max,f}^{n}}{\omega_{f}}}},{f \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & (20)\end{matrix}$and either ω_(f)≤0, the backward wave speeds are given, for each linkf∈E, or they are obtained from a triangular fundamental diagram as afunction of the maximal flow possible and the corresponding centralcritical density, ρ_(c,f):

$\begin{matrix}{{\omega_{f} = \frac{Q_{\max,f}^{n}}{\rho_{c,f} - \rho_{\max,f}}},{f \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu}{N.}}}} & (21)\end{matrix}$

Instead of requiring the four cases of system (2) to be satisfied on thelinks, we can equivalently express the intercell flows on the links asthe minimum of the sending and receiving functions, S and R, as is thecase for the junction flows of (16j) and (16k):q _(i,e) ^(n)=min{R _(i+1) ,S _(i)(g _(v) ^(n))}  (22)for i,j+1 cells on link e with v=head(e).Hierarchical Control

In typical traffic signal control systems, under normal operations,certain junctions are designated as the leader of a pre-defined group ofjunctions sharing certain geographical and flow characteristics, asexemplarily shown in FIG. 2. One of the objectives of defining groups ofjunctions is to permit the calibration of so-called “green waves” inwhich the signals at a group of junctions are coordinated so as tomaximize flow out of the group. The role of the lead junction is toreduce the complexity of the management of the group of junctions to thelocal optimization for a single junction. The parameters determined, forexample, based on local optimization of its flow, are then used for theother junctions in the group. One such set of parameters would be theoffsets between the junctions in the group, another would be the splits,or amount of green time to associate to each phase as a proportion ofthe total cycle time. Lastly, the cycle time itself may be a parameter.

Some systems have multiple levels of this type of hierarchy, in whichthe group leaders are associated and, of those, one is designatedleader, and the process repeats. The objective of this type of localizedand simplified operation was to permit some real-time optimization to beperformed while minimizing drastically the computational burden. Indeed,because typical systems perform local computations, if many junctionsindependently performed local optimization of their own parameters, theresulting solution could be sub-optimal for the system as a whole.

On the other hand, as described in the literature elsewhere, numerousglobal network models have been proposed for traffic control. Suchmodels avoid the sub-optimality issue mentioned above, but on the otherhand are unable to take into account the hierarchy present in thetraffic network and used by operators to handle such flow-basedrequirements as green waves, etc.

Hence, we provide a variation of the control model (16a)-(16o) whichtakes into account the hierarchy present in existing systems whilemaintaining a network-wide optimization of the control variables.

Consider a traffic control system in which there are T hierarchicallevels, where level T represents the lowest level and contains junctionswhich follow the parameter settings of the lead junction of its group,and 1 represents the highest level of the hierarchy. The hierarchy canbe represented as a tree (or a forest) with the lowest level ofjunctions, level T, at the leaves of the tree. The leaves are orderedinto groups of related junctions, one of which is designated as theleader; then the parent of the leaves is defined as the leader of eachgroup, at level T−1. If there are multiple groups each with a leader,then those leaders are coordinated at the next level of the tree, T−2.The tree terminates may terminate with a root node or with multipleparents if no single junction leads the others. In the latter case, wehave a forest, in the former, it is a tree. Note then that the leadjunctions are repeated at multiple levels of the tree or forest.

In order to represent the hierarchy, assign a default weight of 1 toeach leaf node. Then, the weight assigned to the parent of the leaves ineach group of level t is given by

$\begin{matrix}{{w_{v,{t - 1}} = {\sum\limits_{u \in V_{v,t}^{\prime}}\; w_{u,t}}},} & (23)\end{matrix}$where V_(v,t)′ is the group of junctions lead by node v at level t andw_(u,t)=0 if the junction u is not present at level t. As such, theweight for node v at level t−1 is higher than the sum of the weight ofthe non-lead junctions in its group. The process is repeated at eachlevel of the hierarchy.

The weights of the junctions are then summed over all levels of thehierarchy,

$\begin{matrix}{w_{v} = {\sum\limits_{t = 1}^{T}\; w_{v,t}}} & (24)\end{matrix}$The objective function (16a) is defined in terms of flows andspecifically in terms of flows of cells into the junctions, q_(Ce) ^(n).To model the hierarchy on the junctions, we thus modify the weights onthose flow variables as follows:

$\begin{matrix}{{\max\limits_{\rho,q,t,h}{\sum\limits_{n = 1}^{N}\;{\sum\limits_{e \in E}\;{g_{e}^{n}w_{{head}{(e)}}q_{C_{e},e}^{n}}}}} + {\sum\limits_{i = 0}^{C_{e} - 1}\;{\gamma_{ei}q_{i,e}^{n}}}} & (25)\end{matrix}$

FIG. 2 shows a simplistic example 200 of this approach, as exemplarilyapplied to a small portion of a traffice network, if the weights at thedifferent levels are calculated as described above.

Hierarchical Control in the Presence of Incidents

In order to take into account the impact of one or more incidents on thenetwork, this framework can be used with a modification of thedefinition of the lead junction. FIG. 3 shows a simplistic example 300of this modification.

Specifically, define the set Ê^(n) as the set of links with incidentsactive or predicted to be active at time n, and let z_(e) ^(n), ∈[0,1]be the percent capacity reduction due to the incident on link e′ duringtime period n. Further, define the set of junctions having an incidentof high severity on a link outgoing from the junction {circumflex over(V)}^(n)={{circumflex over (v)}^(n); {circumflex over(v)}^(n)=tail(e′),e′∈Ê^(n)}, numbering them in decreasing order ofseverity.

At level T−1 of the tree, assign the first incident-affected junction,{circumflex over (v)}₁, to be a parent node of its group, and so on forthe other incident-affected junctions, unless its parent is anincident-affected junction higher on the severity list. At higher levelsof the tree, the question is which junction is to be designated theleader, and in particular, if the incident-affected junction is toreplace the previously-designated leader.

Incidents themselves can be modeled by reducing the maximal capacity ofthe incident-affected link. Consider e′ an incident-affected link. Fromthe capacity reduction, z_(e) ^(n), the remaining maximal capacity isgiven by the valueQ _(max,e′) ^(n′)=(1−z _(e′) ^(n))Q _(max,e′)  (26)is employed during all such periods n in the model. In addition to thereduction of capacity on the incident-affected link e′, the splittingrates need to be adjusted as well. We assume that for each link of thenetwork there is at least one potential detour route in the event of aninability to traverse the link due to an incident. Thus the splittingrates must be redefined in terms of the available detour routes aroundthe incident-affected link. For all links f predecessors of theincident-affected link e′, p_(fe′) ^(n), f∈pred(e′), the splittingpercentages are redefined as p_(fe) ^(n′),p _(fe′) ^(n′)=(1−z _(e) ^(n))p _(fe′) ^(n).  (27)The remaining quantity z_(e) ^(n)p_(fe′) ^(n), is allocatedproportionally to the non-incident outgoing links from f, and thenoptimized via the use of the control parameters h_(fe′) ^(n), i.e.,

$\begin{matrix}{p_{f\;\overset{\sim}{e}}^{n^{\prime}} = \frac{z_{e}^{n}p_{{fe}^{\prime}}^{n}}{{{O\left( {{head}(f)} \right)}} - 1}} & \left( {28a} \right) \\{h_{f\;\overset{\sim}{e}}^{n} > 0} & \left( {28b} \right)\end{matrix}$for all ({tilde over (e)},f)∈{e∈E:tail(e)=head(f)=tail(e′)}×{f∈E:f∈pred(e′)}.

When an incident disrupts a pre-programmed “green wave”, the controlmodel is designed to generate an alternative corridor that avoids theincident link. This is achieved by the optimization of the controlparameters h just upstream of and around the incident link.

Leveraging Additional Network Data to Reduce Model Complexity

The purpose of this section is twofold. On the one hand, a goal is tofurther constrain the set of controllable resources to be optimized t,h, to those having the most impact on the incident-impacted links.Hence, reducing the feasible plan set accomplishes the objective ofreducing the dimensionality of the optimization problem, as well asensuring that the resources to be controlled are those that are, orwould be, the most heavily saturated from the incident. On the otherend, a goal is to include additional data sources to increase theaccuracy of the resulting control suggestions. To this end weincorporate simulated path flows and observed traffic counts into themodel.

A key input to the definition of the feasible plans is the set oftypical paths and path flows on the network. These are expected to beprovided by a static or dynamic descriptive model of the traffic flow onthe network, such as via a static or dynamic user equilibrium orsimulation model. In many cases, such static or dynamic path and pathflow information will not be available and no reasonable feasible planset reduction can be achieved. However, in cases where input from atraffic assignment module is available, it is of both computational andpractical benefit to make use of the path and flow data to identify thesubset of signals typical most critical to congestion reduction in thepresence of an incident.

Let W⊂V×V be a set of origin-destination (OD) pairings. For each pairingω=(orig(ω),dest(ω)), ω∈W, there is a demand for travel from orig(ω) todest(ω) at time n. Traffic enters the network at orig(ω) at time n,bound for dest(ω), at a rate r_(w) ^(n), where origin nodes areconsidered as sources and destination nodes as sinks. For each node v∈V,we define the sets: W_(O)(v):={ω∈W| orig(ω)=ν} and W₁(v):={ω∈W| dest(ω)=v}. The time-varying OD demands for the network are contained in the|W|-vector r^(n).

Drivers choose a path from their origin to their destination at adeparture time, n. Let P^(n) be the set of possible time-varying pathsthrough the network having departure time n. For each w∈W we define theset:P _(w) ^(n) ⊂P ^(n) :={j∈P:j from orig(w) at time n to dest(w)},

where each path j may be represented as an ordered sequence of 2-tuples:j={(e ₁ ,n ₁);(e ₂ ,n ₂); . . . }

and where (e₁,n₁) refers to the first link on the path j and n₁ the timethat the link is entered, and so on. We relate paths and links through aset of indicator functions, where l_(e) ^(j) is equal to 1 if link e iscontained in path j. We define also {circumflex over (q)}_(je) ^(n) asthe average volume of flow from path j on link e at time n.

Flow Contribution Factor Matrices

The proposed feasible plan reduction approach works as follows. Throughthe use of the path flow input data, one or more flow contributionfactor matrices are computed. These matrices then allow for thecalculation of the set of candidate junctions for traffic control, aswell as determining the form of the potential traffic control to beperformed. Hence, the size of the traffic control is effectively reducedby the limitation of the number of traffic signal control decisionvariables and potentially by the allowable range of each.

A flow contribution factor is defined as Γ_(ef) ^(n), which is thepercentage of flow on link f at time n+T that is on link e at time n,for T=1 . . . N−n. Consider first the case in which a single, static setof paths is available on the network so that the time indices aredropped. Then, the downstream flow contribution factor matrix Γ⁻ can becomputed via a breadth-first-search by storing the following valuesrow-by-row for each link ∈∈E. For all children of link e,f₁={f∈O(head(e))}, Γ_(e,f) ₁ ⁻=p_(e,f) ₁ , where p_(e,f) is, as before,the splitting rate from link e to link f. Then, for the children of thechildren of link e, {f₂}, Γ_(e,f) ₂ ⁻=p_(e,f) ₁ p_(f) ₁ _(,f) ₂=(Γ_(e,f) ₁ ⁻)p_(f) ₁ _(,f) ₂ , so that, in general, the eth row of thedownstream flow contribution factor matrix is defined asΓ_(e,f) ₁ ⁻ =p _(e,f) ₁ ,∀f ₁ ∈O(head(e)),  (29)Γ_(e,f) _(α) ⁻=γ_(e,f) _(α-1) ⁻ p _(f) _(α-1,) _(f) _(α) ∈O(head(f_(α)))  (30)

The upstream contribution factor matrix, Γ⁺ is computed for static pathsets in an analogous manner, withΓ_(e,f) ₁ ⁺ =p _(f) ₁ _(,e) ,∀f ₁ ∈I(tail(e)),  (31)Γ_(e,f) _(α) ⁺=Γ_(e,f) _(α-1) ⁺ p _(f) _(α) _(,f) _(α-1) ,∀f _(α)∈I(tail(f _(α-1)))  (32)

with the eth column in this case defining the most pertinent upstreamlinks f=1, . . . E with respect to the link e.

By keeping track of the cumulative distances from link e to the otherlinks, a threshold dist(e,f)≤γ₁ may be used to limit the depth of thebreadth-first search. Similarly, or in addition, a threshold in terms ofthe magnitude of Γ_(e,f) ⁻≥γ₂ be used to limit the search. For example,a geographically restricted optimized control plan for an incident atlink e starting at time n may include the set of downstream links J_(e)⁻(γ₁, γ₂) such thatJ _(e) ⁻(γ₁,γ₂)={f:γ _(e,f) ⁻≥γ₁,dist(e,f)≤γ₂ ,∀f∈E}.  (33)The definition of J_(e) ⁺ is analogous with the difference being only asubstitution of the vector Γ_(e) ⁺. By varying the parameters γ₁ and γ₂,a family of feasible plan sets may be computed.

While in many cases, only static paths on the network are available, itis desirable when possible to make use of time-varying paths. Iftime-varying path sets are available, then multiple contribution factormatrices may be computed, with each defined in terms of the departuretime of the path set.

Thus, for a finite horizon of length N, up to 2N such matrices may becomputed, one downstream and one upstream matrix for each desireddeparture period, Γ^(−,n) and Γ^(+,n), respectively. Defining the subsetof paths that enter link e at time n as {tilde over (P)}_(e) ^(n), wecompute the number of upstream vehicles entering link f which will enterlink e at time n, {tilde over (q)}_(fe) ^(n), by

$\begin{matrix}{q_{fe}^{- n} = {\sum\limits_{j \in {\overset{\sim}{p}}_{e}^{n}}{\sum\limits_{n^{\prime} \geq n}{{{\hat{q}}_{fe}\left( n^{\prime} \right)}.}}}} & (34)\end{matrix}$

Note that the subset P_(e) ^(n) defines the set of paths departing fromlink e at time n whereas the set {tilde over (P)}_(e) ^(n) requiredabove is composed of those paths traversing link e at time n.

Similarly, the number of downstream vehicles entering link f havingentered link n at time n, {tilde over (q)}_(ijt){tilde over (q)}_(ef)^(n) is

$\begin{matrix}{{\overset{\sim}{q}}_{ef}^{n} = {\sum\limits_{j \in {\overset{\sim}{p}}_{e}^{n}}{\sum\limits_{n^{\prime} \geq n}{{{\hat{q}}_{ef}^{n}\left( n^{\prime} \right)}.}}}} & (35)\end{matrix}$Then,

$\begin{matrix}{{\Gamma_{ef}^{- {,n}} = \frac{{\overset{\sim}{q}}_{ef}^{n}}{\sum\limits_{j \in p_{e}^{n}}{\hat{q}}_{kj}^{n^{\prime}}}},} & (36) \\{\Gamma_{ef}^{+ {,n}} = {\frac{{\overset{\sim}{q}}_{ef}^{n}}{\sum\limits_{j \in p_{e}^{n}}{{\hat{q}}_{kj}\left( t^{\prime} \right)}}.}} & (37)\end{matrix}$In this case, the geographically restricted, time-varying set ofjunctions to include in a control plan optimization are defined by a setof sets. Given an incident at link e starting at time n and of expectedduration κ_(e) ^(n), the set of potential downstream links J_(i)^(−,n,n′) (γ₁,γ₂) to include in a control optimization at time n′ isJ _(e) ^(−,n,n′)(γ₁,γ₂)={f:Γ _(e,f) ^(−,n′)≥γ₁,dist(e,f)≤γ₂,∀f∈E},  (38)for n′∈[n,n+κ_(e) ^(n)].

As before, the definition of J_(e) ⁺ is analogous, with the differencebeing only a substitution of the vector Γ_(e′) ^(+,n′). Also as before,by varying the parameters γ1 and γ₂, a family of feasible plan sets maybe computed, in this case for each time step n′.

The Control Model Over the Reduced Plan Set

For a given pair of parameters, (γ₁, γ₂), the reduced set of junctionsto be considered for control optimization can be identified as describedabove. Recall that the set {tilde over (V)}^(n) is all such junctions attime n; add to that set the incident junctions themselves, {circumflexover (V)}^(n), defined in a previous section above as those junctionsdirectly upstream of incident links, so that {circumflex over (V)}^(n)∈{tilde over (V)}^(n). Then, the optimal traffic control problem can bemodified to include as decision variables only those t_(v) ^(n) andh_(v) ^(n) for v∈{tilde over (V)}^(n). Similarly, the set of linksentering junctions {tilde over (V)} is referred to as {tilde over(E)}^(n) at time n which includes as a subset Ê^(n)⊂{tilde over (E)}^(n)the incidents active or predicted to be active at n.

$\begin{matrix}{{\max\limits_{\rho,q,t,h}z} = {\sum\limits_{n = 1}^{N}\left( {{\sum\limits_{e \in E}{{g_{e}^{n}\left( t_{v}^{n} \right)}w_{{head}{(e)}}q_{{Ce},e}^{n}}} + {\sum\limits_{i = 0}^{C_{e} - 1}{ϛ_{ei}q_{i,e}^{n}}}} \right.}} & \left( {39a} \right)\end{matrix}$Such that:

$\begin{matrix}{{\rho_{j,e}^{n + 1} = {\rho_{j,e}^{n} + {\frac{\Delta\; N}{\Delta\; x}\left( {q_{j,e}^{n} - q_{{j - 1},e}^{n}} \right)}}},{j = {1\mspace{14mu}\ldots\mspace{14mu} C_{e}}},{e \in E},{n = 1},{\ldots\mspace{14mu} N}} & \left( {39b} \right) \\{{q_{0,e}^{n} = {\sum\limits_{j \in {I{({{tail}{(e)}})}}}{p_{j,e}^{n}q_{C_{i,j}}^{n}}}},{e \in {E\backslash{\overset{\sim}{E}}^{n}}},{n = 1},{\ldots\mspace{14mu} N}} & \left( {39c} \right) \\{{q_{0,e}^{n} = {\sum\limits_{j \in {I{({{tail}{(e)}})}}}{{p_{j,e}^{n}\left( {t_{v}^{n},h^{n}} \right)}q_{C_{i,j}}^{n}}}},{e \in {\overset{\sim}{E}}^{n}},{n = 1},{\ldots\mspace{14mu} N}} & \left( {39d} \right) \\{{g_{e}^{n} = {\frac{1}{L_{v}}{\sum\limits_{j \in H_{v}}{t_{vj}^{n}{\sum\limits_{{f \in {j{(1)}}}:{e \in {j{(0)}}}}\frac{\eta_{ef}}{Q_{\max.e}}}}}}},{e \in {\overset{\sim}{E}}^{n}},{n = 1},{\ldots\mspace{14mu} N}} & \left( {39e} \right) \\{{h_{ef}^{n + 1} = h_{ef}^{n}},{{\left( {n + 1} \right)\;{mod}\;\Delta\;\tau} \neq 0},{f \in {O\left( {{head}(e)} \right)}},{e \in {\overset{\sim}{E}}^{n}},{n = 1},{\ldots\mspace{14mu} N}} & \left( {39f} \right) \\{{g_{ef}^{n + 1} = g_{e}^{n}},{{\left( {n + 1} \right)\;{mod}\;\Delta\;\tau} \neq 0},{f \in {O\left( {{head}(e)} \right)}},{e \in {\overset{\sim}{E}}^{n}},{n = 1},{\ldots\mspace{14mu} N}} & \left( {39g} \right) \\{{q_{{i + 1},e}^{n} = {Q_{i,e}\left( \rho^{n} \right)}},{i = 0},{{\ldots\mspace{14mu} C_{e}} - 2},{e \in E},{v = {{head}(e)}},{n = 1},{\ldots\mspace{14mu} N}} & \left( {39h} \right) \\{{q_{C_{e},e}^{n} = {Q_{C_{e},e}\left( {\rho^{n},g_{e}^{n}} \right)}},{e \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {39i} \right) \\{{{\sum\limits_{j \in H_{v}}t_{vj}^{n}} = L_{v}},{v \in \overset{\sim}{V}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {39j} \right) \\{{{\sum\limits_{e \in {I{(v)}}}{B_{{ef},v}q_{C_{e},e}^{n}}} \leq R_{f}^{n}},{f \in {{O(v)}e} \in {E\backslash{\overset{\sim}{E}}^{n}}},{v = {{head}(e)}},{n = 1},{\ldots\mspace{14mu} N}} & \left( {390k} \right) \\{{{\sum\limits_{e \in {I{(v)}}}{{B_{{ef},v}\left( {p^{n}\left( {t_{v}^{n},h^{n}} \right)} \right)}q_{C_{e},e}^{n}}} \leq R_{f}^{n}},{f \in {O(v)}},{e \in {\overset{\sim}{E}}^{n}},{v = {{head}(e)}},{w = {{head}(f)}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {39l} \right) \\{{0 \leq q_{C_{e},e}^{n} \leq S_{e}^{n}},{e \in {E\backslash{\overset{\sim}{E}}^{n}}},{v = {{head}(e)}},{n = 1},{\ldots\mspace{14mu} N}} & \left( {39m} \right) \\{{0 \leq q_{C_{e},e}^{n} \leq {S_{e}\left( g_{v}^{n} \right)}},{e \in {\overset{\sim}{E}}^{n}},{v = {{head}(e)}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {39n} \right) \\{{l_{vj} \leq t_{vj}^{n} \leq u_{vj}},{j \in H_{v}},{v \in \overset{\sim}{V}},{n = 1},{\ldots\mspace{14mu} N}} & \left( {39o} \right) \\{{{- \frac{p_{ef}^{n}\left( t_{v}^{n} \right)}{\xi_{ef}^{n}}} \leq h_{ef}^{n} \leq \frac{1 - {p_{ef}^{n}\left( t_{v}^{n} \right)}}{\xi_{ef}^{n}}},{f \in {O\left( {{head}(e)} \right)}},{e \in {\overset{\sim}{E}}^{n}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {339p} \right) \\{{0 \leq \rho_{i}^{n} \leq \rho_{\max,e}},{i = 0},{\ldots\mspace{14mu} C_{e}},{e \in E},{v = 1},{\ldots\mspace{14mu} N}} & \left( {39q} \right) \\{{q_{i,e}^{n} \geq 0},{i = 1},{\ldots\mspace{14mu} C_{e}},{e \in E},{n = 1},{\ldots\mspace{14mu} N}} & \left( {39r} \right)\end{matrix}$Decomposition-Based Simulation-Optimization Heuristic

The very large size of the model (39a)-(39r) for networks of evenmoderate size means that many techniques typically used on nonlinearprogramming models are not effective in this case. For that reason, wedevelop a decomposition-based simulation-optimization method that takesadvantage of the effective algorithms available for solving the networkflow of subproblem. For a fixed set of control variables t and h, theremaining problem in the flow and density variables q and ρ can besolved via forward simulation-type methods which are highly efficient,even on problems a medium to large size. Conversely, given a flow basedon a fixed set of controls t and h, the control variables can be updatedby solving an optimization problem, but in far fewer variables andconstraints.

Simulation Sub-Problem

The idea behind the method is, therefore, to start with an initial setof values for the control variables t^(k) and h^(k), and henceg^(k)(t^(k)), for iteration k=0, and to solve the macroscopic networkflow problem given by, for each cell i on link e and over all links e∈E:

$\begin{matrix}{{\rho_{i}^{n + 1} = {\rho_{i}^{n} + {\frac{\Delta\; N}{\Delta\; x}\left( {{Q\left( \rho_{i}^{n} \right)} - {Q\left( \rho_{j}^{n} \right)}} \right)}}},} & (40)\end{matrix}$with, as before,

$\begin{matrix}{Q_{i,{i + 1}}^{k,n} = \left\{ \begin{matrix}{{Q\left( \rho_{i}^{n} \right)},} & {{if}\mspace{14mu}\left( {\rho_{j}^{n} \leq \rho_{i}^{n} \leq \rho_{c}} \right)} \\{{Q\left( \rho_{i + 1}^{n} \right)},} & {{if}\mspace{14mu}\left( {\rho_{c} \leq \rho_{i + 1}^{n} \leq \rho_{i}^{n}} \right)} \\{Q_{\max,e}^{k,n},} & {{if}\mspace{14mu}\left( {\rho_{i + 1}^{n} \leq \rho_{c} \leq \rho_{i}^{n}} \right)} \\{{\min\left\{ {{Q\left( \rho_{i}^{n} \right)},{Q\left( \rho_{i + 1}^{n} \right)}} \right\}},} & {{if}\mspace{14mu}\left( {\rho_{i}^{n} \leq \rho_{i + 1}^{n}} \right)}\end{matrix} \right.} & (41)\end{matrix}$where this cell C_(e) is on link e, and if e∈{tilde over (E)}^(n) fortime step n,

$\begin{matrix}{{{Q_{\max,e}^{k}\left( \rho_{c,e} \right)} = {{Q\left( \rho_{c,e} \right)}\frac{\sum\limits_{f \in {O{({{head}{(e)}})}}}{g_{e}^{k,n}\left( t_{v}^{k,n} \right)}}{{O\left( {{head}(e)} \right)}}}},} & (42)\end{matrix}$Otherwise, if e∈E\{tilde over (E)}^(n) at time step n, Q_(max,e) ^(k,n)is a constant independent of the control variables t.

At the junctions, v∈V\{tilde over (V)}^(n), the original junction modelof equations (4)-(6) is solved, whereas the following linear program issolved with fixed values of g^(k,n)=g^(k,n)(t^(k,n)) and h^(k,n) forjunctions v∈{tilde over (V)}^(n):

$\begin{matrix}{\mspace{79mu}{{{q^{*}\left( {g^{k},{h^{k}\left( t^{k} \right)}} \right)} \in {\arg\underset{\{ q_{e}\}}{\;\max}{\sum\limits_{e \in {I{(v)}}}q_{e}}}}\mspace{79mu}{{such}\mspace{14mu}{that}}\mspace{79mu}{{{{B_{v}\left( {p\left( h^{k,n} \right)} \right)}q} \leq R_{f}^{k,n}},{f \in {O(v)}},\mspace{79mu}{0 \leq q \leq {S_{e}^{k,n}\left( g_{{head}{(e)}}^{k,n} \right)}},{e \in {I(v)}},}}} & (43) \\{{S_{e}^{k,n}\left( {g_{{head}{(e)}}^{n}\left( t_{v}^{k,n} \right)} \right)} = \left\{ \begin{matrix}{{v_{e}\rho_{e}^{n}},} & \left( {{{if}\mspace{14mu}\rho_{e}} \in \left\lbrack {0,\rho_{c_{1},e}} \right\rbrack} \right. \\{{Q_{\max,e}^{n}{g_{{head}{(e)}}^{k,n}\left( t_{v}^{k,n} \right)}},} & \left( {{{if}\mspace{14mu}\rho_{e}} \in {{\rho_{c_{1},e},\rho_{\max,f}}}} \right. \\{0,} & {otherwise}\end{matrix} \right.} & (44) \\{\mspace{79mu}{and}} & \; \\{\mspace{79mu}{R_{f}^{n} = \left\{ \begin{matrix}{Q_{\max,f^{\prime}}^{n},} & \left( {{{if}\mspace{14mu}\rho_{j}} \in \left\lbrack {0,\rho_{c_{2},f}} \right\rbrack} \right) \\{{Q_{\max,f}^{n} + {\omega_{f}\left( {\rho_{f}^{n} - \rho_{c_{2},f}^{n}} \right)}},} & \left( {{{if}\mspace{14mu}\rho_{f}} \in \left\lbrack {\rho_{c_{2},f},\rho_{\max,f}} \right\rbrack} \right. \\{0,} & {otherwise}\end{matrix} \right.}} & (45) \\{\mspace{79mu}{where}} & \; \\{\mspace{79mu}{{\rho_{c_{1},e}^{k,n} = \frac{Q_{\max,e}^{n}g_{{head}{(e)}}^{n}}{v_{e}}},\mspace{14mu}{e \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}}} & (46) \\{\mspace{79mu}{{\rho_{c_{2},f}^{k,n} = {\rho_{\max,f} + \frac{Q_{\max,f}^{n}}{\omega_{f}}}},\mspace{14mu}{f \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}}} & (47)\end{matrix}$with, as before, either ω_(f)≤0, the backward wave speeds, are given foreach link f∈E, or they are obtained from a triangular fundamentaldiagram as a function of the maximal flow possible and the correspondingcentral critical density:

$\begin{matrix}{{\omega_{f} = \frac{Q_{\max,f}^{n}}{\rho_{c,f} - \rho_{\max,f}}},\mspace{14mu}{f \in E},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & (48)\end{matrix}$Optimization Sub-Problem

Then, given the current flow solution q^(k)=q*(t^(k),h^(k)) thefollowing nonlinear programming problem is solved, to obtaint^(k+1)(q^(k)), h^(k+1)(q^(k)), where p_(ef) ^(n)(t_(v) ^(n),h^(n)) isnonlinear in t_(v) ^(n):

$\begin{matrix}{{\max\limits_{t,h}Z} = {\underset{n = 1}{\overset{N}{\sum\quad}}\left\lbrack {{\underset{e \in \overset{\sim}{E}}{\sum}\left( {{{g_{e}^{n}\left( t_{v}^{n} \right)}w_{{head}{(e)}}q_{C_{e},e}^{k,n}} + {\sum\limits_{j \in {I{({{tail}{(e)}})}}}{{p_{j,e}^{n}\left( h^{n} \right)}q_{C_{j},j}^{k,n}}}} \right)} + {\sum\limits_{i = 0}^{C_{e} - 1}{\zeta_{ei}q_{ie}^{k,n}}}} \right\rbrack}} & \left( {49a} \right)\end{matrix}$Such that:

$\begin{matrix}{{q_{0,e}^{k,n} = {\sum\limits_{j \in {I{({{tail}{(e)}})}}}{{\rho_{j,e}^{n}\left( {t_{v}^{n},h^{n}} \right)}q_{{Cj},j}^{k,n}}}},{e \in {\overset{\sim}{E}}^{n}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {49b} \right) \\{{g_{e}^{n} = {\frac{1}{L_{v}}{\sum\limits_{j \in H_{v}}{t_{vj}^{n}{\sum\limits_{{f \in {j{(q)}}}:{e \in {j{(0)}}}}\frac{\eta_{ef}}{Q_{\max,e}}}}}}},{e \in \overset{\sim}{E}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {49c} \right) \\{{h_{ef}^{n + 1} = h_{ef}^{n}},{{\left( {n + 1} \right)\;{mod}\;\Delta\;\tau} \neq 0},{f \in {O\left( {{head}(e)} \right)}},{e \in \overset{\sim}{E}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {49d} \right) \\{{{{g_{e}^{n + 1}\left( t_{v}^{n} \right)} = {g_{e}^{n}\left( t_{v}^{n} \right)}},{{\left( {n + 1} \right){mod}\;\Delta\;\tau} \neq 0},{f \in {O\left( {{head}(e)} \right)}},{e \in \overset{\sim}{E}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}}{{{\sum\limits_{j \in H_{v}}t_{vj}^{n}} = L_{v}},{v \in \overset{\sim}{V}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}}} & \left( {49e} \right) \\{{{\sum\limits_{e \in {I{(v)}}}{{B_{{ef},v}\left( {p^{n}\left( {t_{v}^{n},h^{n}} \right)} \right)}q_{C_{e},e}^{k,n}}} \leq R_{f}^{n}},{f \in {O(v)}},{e \in {\overset{\sim}{E}}^{n}},{v = {{head}(e)}},{w = {{head}(f)}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {49f} \right) \\{{0 \leq q_{C_{e},e}^{k,n} \leq {S_{e}\left( {g_{e}^{n}\left( t_{v}^{n} \right)} \right)}},{e \in {\overset{\sim}{E}}^{n}},{v = {{head}(e)}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {5490h} \right) \\{{l_{vj} \leq t_{vj}^{n} \leq u_{vj}},{j \in H_{v}},{v \in \overset{\sim}{V}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & (49) \\{{{- \frac{p_{ef}^{n}\left( t_{v}^{n} \right)}{\xi_{ef}^{n}}} \leq h_{ef}^{n} \leq \frac{1 - {p_{ef}^{n}\left( t_{v}^{n} \right)}}{\xi_{ef}^{n}}},{f \in {O\left( {{head}(e)} \right)}},{e \in {\overset{\sim}{E}}^{n}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}} & \left( {49j} \right)\end{matrix}$Linearized Optimization Sub-Problem

In this section, we consider a simplified version of the optimizationsub-problem in which the split rates at junctions, e.g., equation 11, donot depend upon the traffic control variables, t. In addition, some ofthe flow-based constraints on the control variables are relaxed andhandled via the simulation sub-problem only.

$\begin{matrix}{{p_{ef}^{n} = \frac{\eta_{ef}}{\sum\limits_{j \in H_{v}}{\sum\limits_{{({e,f})} \in j}\eta_{ef}}}},} & (50)\end{matrix}$for each pair (e,f) such that head(e)=tail(f)=v. Recall that η_(ef) ^(j)is the portion of the capacity of link e pertaining to the movement ef,such as the capacity of a turning lane if the movement ef is a turningmovement.

Then, the decomposed simulation-optimization problem of (49a)-(49j)reduces to the following linear program, for given (q^(k),ρ^(k)).

$\begin{matrix}{{\max\limits_{t}Z} = {\quad{\sum\limits_{n = 1}^{N}\left\lbrack {{\sum\limits_{e \in \overset{\sim}{E}}\left( {{{g_{e}^{n}\left( t_{v}^{n} \right)}w_{{head}{(e)}}q_{C_{e},e}^{k,n}} + {\sum\limits_{j \in {I{({{tail}{(e)}})}}}{{p_{j,e}^{n}\left( h^{n} \right)}q_{C_{j},j}^{k,n}}}} \right)} + \left. \quad{\sum\limits_{i = 0}^{C_{e} - 1}{\varsigma_{ei}q_{ie}^{k,n}}} \right\rbrack} \right.}}} & \left( {51a} \right) \\{\mspace{85mu}{{such}\mspace{14mu}{that}}} & \; \\{\mspace{79mu}{{g_{e}^{n} = {\frac{1}{L_{v}}{\sum\limits_{j \in H_{v}}{t_{vj}^{n}{\sum\limits_{{f \in {j{(1)}}}:{e \in {j{(0)}}}}\frac{\eta_{ef}}{Q_{\max,e}}}}}}},{e \in {\overset{\sim}{E}}^{n}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}}} & \left( {51b} \right) \\{\mspace{79mu}{{{g_{e}^{n + 1}\left( t_{v}^{n} \right)} = {g_{e}^{n}\left( t_{v}^{n} \right)}},{{\left( {n + 1} \right){mod}\;\Delta\;\tau} \neq 0},{f \in {O\left( {{head}(e)} \right)}},\mspace{79mu}{e \in {\overset{\sim}{E}}^{n}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}}} & \left( {51c} \right) \\{\mspace{79mu}{{{\sum\limits_{j \in H_{v}}t_{vj}^{n}} = L_{v}},\mspace{11mu}{v \in \overset{\sim}{V}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}}} & \left( {51d} \right) \\{\mspace{79mu}{{0 \leq q_{C_{e},e}^{k,n} \leq {S_{e}\left( {g_{e}^{n}\left( t_{v}^{n} \right)} \right)}},{e \in {\overset{\sim}{E}}^{n}},{v = {{head}(e)}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}}} & \left( {51e} \right) \\{\mspace{79mu}{{l_{vj} \leq t_{vj}^{n} \leq u_{vj}},{j \in H_{v}},{v \in \overset{\sim}{V}},{n = {1\mspace{14mu}\ldots\mspace{14mu} N}}}} & \left( {52f} \right)\end{matrix}$Exemplary Hardware Implementation

FIG. 6 illustrates a typical hardware configuration of an informationhandling/computer system in accordance with the invention and whichpreferably has at least one processor or central processing unit (CPU)611.

The CPUs 611 are interconnected via a system bus 612 to a random accessmemory (RAM) 614, read-only memory (ROM) 616, input/output (I/O) adapter618 (for connecting peripheral devices such as disk units 621 and tapedrives 640 to the bus 612), user interface adapter 622 (for connecting akeyboard 624, mouse 626, speaker 628, microphone 632, and/or other userinterface device to the bus 612), a communication adapter 634 forconnecting an information handling system to a data processing network,the Internet, an Intranet, a personal area network (PAN), etc., and adisplay adapter 536 for connecting the bus 612 to a display device 638and/or printer 639 (e.g., a digital printer or the like).

In addition to the hardware/software environment described above, adifferent aspect of the invention includes a computer-implemented methodfor performing the above method. As an example, this method may beimplemented in the particular environment discussed above.

Such a method may be implemented, for example, by operating a computer,as embodied by a digital data processing apparatus, to execute asequence of machine-readable instructions. These instructions may residein various types of signal-bearing storage media.

Thus, this aspect of the present invention is directed to a programmedproduct, comprising non-transitory, signal-bearing storage mediatangibly embodying a program of machine-readable instructions executableby a digital data processor incorporating the CPU 611 and hardwareabove, to perform the method of the invention.

This non-transitory signal-bearing storage media may include, forexample, a RAM contained within the CPU 611, as represented by thefast-access storage for example. Alternatively, the instructions may becontained in another non-transitory signal-bearing storage media, suchas a magnetic data storage diskette 700 (FIG. 7), directly or indirectlyaccessible by the CPU 711.

Whether contained in the diskette 700, the computer/CPU 611, orelsewhere, the instructions may be stored on a variety ofmachine-readable data storage media, such as DASD storage (e.g., aconventional “hard drive” or a RAID array), magnetic tape, electronicread-only memory (e.g., ROM, EPROM, or EEPROM), an optical storagedevice (e.g. CD-ROM, WORM, DVD, digital optical tape, etc.), paper“punch” cards, or other suitable signal-bearing storage media, includingstorage devices in transmission media involving either digital oranalog, communication links, and wireless. In an illustrative embodimentof the invention, the machine-readable instructions may comprisesoftware object code.

While the invention has been described in terms of various preferredembodiments, those skilled in the art will recognize that the inventioncan be practiced with modification within the spirit and scope of theappended claims.

Further, it is noted that, Applicants' intent is to encompassequivalents of all claim elements, even if amended later duringprosecution.

Having thus described our invention, what we claim as new and desire tosecure by Letters Patent is as follows:
 1. A method for adjustingcontrol parameters of a traffic management system in a presence of oneor more incidents on a network, said method comprising: representingnetwork junctions in a tree format, wherein junctions at a lowest levelof a tree comprise a group of associated junctions whose timings arecoordinated to maximize a traffic flow out of the group, wherein eachjunction is associated with a weight as a function of its height withinits tree, and wherein a junction with a highest weight in each treecomprises a lead junction for receiving timing information for its groupof associated junctions; and solving, as executed by a processor on acomputer, a real-time optimization of control parameters for thenetwork, using the weights of the junctions, wherein, upon an occurrenceof an incident in the network and depending upon a severity level of theincident, an incident-affected junction is selectively changed inelevation within its tree, and reallocated junction weights resultingfrom the incident are used for solving the optimization of networkcontrol parameters.
 2. The method of claim 1, further comprisingproviding an output to change one or more control parameters in thenetwork.
 3. The method of claim 2, wherein the network describes a roadtransportation network.
 4. The method of claim 3, wherein the controlparameters are changed by at least one of: a hard control, whereby atraffic timing is changed; and a soft control, whereby at least one ofinformation and a recommendation is displayed to drivers.
 5. The methodof claim 3, further comprising: exercising a traffic simulation model,to identify typical routes of drivers; and identifying junctions in saidnetwork most likely to require traffic control to alleviate anincident-related congestion.
 6. The method of claim 3, whereinprioritized corridors, “green waves”, are defined in said network,wherein traffic signals for a series of junctions in said network arecoordinated to permit waves of traffic to flow in said series ofjunctions.
 7. The method of claim 6, wherein, in a presence of anincident, traffic is selectively diverted from one or more affectedgreen wave through an alternate corridor of said network.
 8. The methodof claim 2, wherein optimal control values are computed a posteriori andserve as starting points when similar incidents occur or for futureplanned events.
 9. The method of claim 2, wherein the network describesone of an energy grid network and a water supply network.
 10. The methodof claim 1, further comprising: receiving data indicating currenttraffic in the network; and detecting that an incident has occurred. 11.The method of claim 10, wherein said detecting an incident has occurredcomprises at least one of: using the current traffic data toautomatically detect the incident; receiving an input that indicates theincident has occurred; and receiving an input that indicates that aplanned event will invoke the incident in the network.
 12. The method ofclaim 1, wherein the real-time optimization comprises using a trafficmodel comprising a macroscopic fluid flow model, describing flow oftraffic on links in said network, and a junction model, describing howtraffic on an incoming link to a node of said network is propagated tooutgoing links of said node.
 13. The method of claim 12, wherein anincident is modeled by reducing a maximal capacity of anincident-affected link.
 14. The method of claim 1, as incorporated intoa control optimization framework that solves a time-dependent controlproblem on a rolling-horizon framework, implementing only the firstcontrol point and then updating time steps and solving again.
 15. Anon-transitory, computer-readable storage medium tangibly embodying aset of machine-readable instructions to execute the method of claim 1.16. An apparatus, comprising: a central processing unit (CPU); and amemory, wherein said memory has tangibly embodied thereon a set ofmachine-readable instructions for executing a method for adjustingcontrol parameters of a traffic management system in a presence of oneor more incidents on a network, said method comprising: representingnetwork junctions in a tree format, wherein junctions at a lowest levelof a tree comprise a group of associated junctions whose timings arecoordinated to maximize a traffic flow out of the group, wherein eachjunction is associated with a weight as a function of its height withinits tree, and wherein a junction with a highest weight in each treecomprises a lead junction for receiving timing information for its groupof associated junctions; and solving a real-time optimization of controlparameters for the network, using the weights on the junctions, wherein,upon occurrence of an incident in the network and depending upon aseverity level of the incident, an incident-affected junction isselectively changed in elevation within its tree, and reallocatedjunction weights resulting from the incident are used for solving theoptimization of network control parameters.
 17. The apparatus of claim16, wherein the network comprises a road transportation network, saidapparatus further comprising: an input port to receive current trafficdata; and an output port to provide output signals controlling one ormore of the network control parameters.
 18. The apparatus of claim 17,wherein the output signals comprise at least one of: a hard controlsignal, whereby timings for traffic lights in the network arecontrolled; and a soft control, whereby a signal providing at least oneof information and a recommendation is provided to drivers in thenetwork.
 19. A method of traffic control, comprising: representing anetwork for traffic as comprising a plurality of links interconnected bya plurality of nodes, wherein network junctions are represented in atree format with junctions at a lowest level of a tree comprising agroup of associated junctions whose timings are coordinated to maximizea traffic flow out of the group, wherein each junction is associatedwith a weight as a function of its height within its tree, and wherein ajunction with a highest weight in each tree comprises a lead junctionfor receiving timing information for its group of associated junctions;loading a traffic model for execution by a processor on a computer, thetraffic model comprising a fluid model, that describes a flow of trafficwithin links of said network, and a junction model, that describes howtraffic on an incoming link to a node is propagated to outgoing links ofthe node, the traffic model assigning priority weights to the nodes ofthe network using weights of the tree format and using the priorityweights for calculating parameters for optimal traffic control in thenetwork; receiving, as input data into the computer, indication ofcurrent traffic on the network; exercising the traffic model with thereceived input data to calculate one or more control parameters; andoutputting the one or more control parameters from the computer to oneor more devices in the network that control traffic flow, wherein, uponan occurrence of an incident that causes a disruption in an existingtraffic flow on the network, the traffic model adapts to the incident bychanging one or more of the priority weights of nodes within thenetwork, depending upon a location in the network and a severity of theincident.
 20. The method of claim 19, wherein the traffic model:includes an arrangement of nodes of the network in a hierarchy of nodes;assigns a priority weight to each node based upon its level location inthe hierarchy of nodes; and adapts to incidents by modifying thehierarchy of nodes to place nodes affected by the incident at a higherlevel in the hierarchy of nodes, thereby providing different weights tonodes affected by the incident and permitting the traffic model tocalculate values for the control parameters that automatically adapt tothe incident.